Real algebraic curves on real del Pezzo...

Real algebraic curves on real del Pezzo surfaces

Matilde Manzaroli

Abstract

The study of the topology of real algebraic varieties dates back to the work ofHarnack, Klein and Hilbert in the 19th century; in particular, the isotopy typeclassification of real algebraic curves in real toric surfaces is a classical subjectthat has undergone considerable evolution. On the other hand, not much isknown for more general ambient surfaces. We take a step forward in the studyof topological type classification of real algebraic curves on non-toric surfacesfocusing on real del Pezzo surfaces of degree 1 and 2 with multi-componentsreal part. We use degeneration methods and real enumerative geometry incombination with variations of classical methods to give obstructions to theexistence of topological type classes realised by real algebraic curves and to giveconstructions of real algebraic curves with prescribed topology.

Contents1 Introduction 2

1.1 Classification and generalities . . . . . . . . . . . . . . . . . . . . . . . 21.2 k-sphere real degree 2 del Pezzo surfaces . . . . . . . . . . . . . . . . . 31.3 k-sphere real degree 1 del Pezzo surfaces . . . . . . . . . . . . . . . . . 5

2 Preliminaries 52.1 Encoding topological types and real schemes . . . . . . . . . . . . . . . 52.2 Hirzebruch surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Dessins d’enfants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Real curves on k-sphere real del Pezzo surfaces of degree 2 93.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Real schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Obstructions and Welschinger-type invariants . . . . . . . . . . . . . . 133.5 Class 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Class 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7 Symplectic curve on a 4-sphere real symplectic degree 2 del Pezzo surface 173.8 Constructions via patchworking . . . . . . . . . . . . . . . . . . . . . . 17

3.8.1 Intermediate constructions: Constructions on quadric surfacesand on j-sphere real 1-nodal del Pezzo pairs of degree 2 . . . . 20

3.9 Final constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Real curves on k-sphere real del Pezzo surfaces of degree 1 314.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 J -Obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Real schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Positive and negative connected components . . . . . . . . . . . . . . . 334.5 Bézout-type obstrction, d ≥ 4 . . . . . . . . . . . . . . . . . . . . . . . 344.6 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.7 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.7.1 Moving hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . 354.7.2 Harnack’s construction method on Q . . . . . . . . . . . . . . . 364.7.3 Five particular constructions and Class 3 . . . . . . . . . . . . 36

1

1 IntroductionThe study of topology of real algebraic varieties dates back to the work of Harnack,Klein and Hilbert in the 19-th century ([Har76], [Hil02], [Kle73]). A real algebraic va-riety (X,σ) is a compact complex algebraic variety equipped with an anti-holomorphicinvolution σ : X → X, called real structure. The real part RX of (X,σ) is the setof points fixed by the involution σ. Hilbert proposed in the first part of his 16-thproblem to classify the isotopy types of real algebraic curves of degree 6 in RP 2,respectively of real algebraic surfaces of degree 4 in RP 3. The classification on RP 2had been achieved by Gudkov ([Gud69]) at the end of the 60’s and the other one byKharlamov ([Kha76], [Kha78]) around ten years later. At the moment, the classifica-tion of real algebraic surfaces in the real projective space is known only up to degree4, respectively that of real algebraic plane curves up to degree 7 ([Vir84a], [Vir84b]).There are two main directions in the study of the topology of real algebraic varieties.The first is to give obstructions to the existence of topological type classes realised byreal algebraic varieties. The second direction is to provide constructions of real alge-braic varieties with prescribed topology. From the 70’s, especially thanks to the workof Arnold and Rokhlin ([Arn71], [Rok72], [Rok74], [Rok78], [Rok80]), many generalobstructions had been discovered. On the other hand, the construction techniqueshad remained relatively elementary for a long time. In 1979, Viro provided a break-through in the construction direction by inventing the patchworking method ([Vir84a],[Vir84b]). Such method and its generalizations still remain the most powerful tools toconstruct real algebraic hypersurfaces with prescribed topology in real algebraic toricvarieties with the standard real structure. Exploiting the patchworking technique,several construction results have been achieved on real algebraic toric varieties. Onthe other hand, only few works have been devoted to classification problems on realnon-toric varieties ([Mik98]) or on toric varieties with the non-standard real structure(for example, [GS80], [Mik94], [DZ99], [Man20]).

Definition 1.1. Let (X,σ) be a real del Pezzo surface. We say that X is a k-spherereal del Pezzo surface if:

(i) X has degree 2 and RX is homeomorphic to⊔kj=1 S

2, with 1 ≤ k ≤ 4 ;

(ii) X has degree 1 and RX is homeomorphic to RP 2 t⊔kj=1 S

2, with 0 ≤ k ≤ 4.

We take a step forward in the classification of embedded topology of real algebraiccurves on real non-toric surfaces studying the topological type classifications of realalgebraic curves on k-sphere real del Pezzo surfaces of degree 1 and 2.In 1998, Mikhalkin ([Mik98]) was the first to face the issue of classifying real algebraiccurves on real algebraic surfaces with non-connected real part. In particular, hestudied the topology of the real part of transverse intersections of real quadric surfaceswith real cubic surfaces in CP 3 equipped with the standard real structure. To ourknowledge, there are no other classifications on real algebraic surfaces with non-connected real part.

1.1 Classification and generalitiesWe look at real curves in real k-sphere del Pezzo surfaces whose homology class isdetermined by an integer, called class.

Definition 1.2. Let (X,σ) be a real del Pezzo surface and let A ⊂ X be a realalgebraic curve. Then, we say that A has class d on X if A realizes dc1(X) inH2(X;Z), where c1(X) is the anti-canonical class of X.

Definition 1.3. Let⊔li=1Bi and

⊔li=1B

′i be two collection of l circles embedded in

the disjoint union V of k spheres (and a real projective plane).

• We say that the pairs (V,⊔li=1Bi) and (V,

⊔li=1B

′i) are homeomorphic if there

exists a homeomorphism f : V → V such that f(⊔li=1Bi) =

⊔li=1B

′i.

• We call topological type any arrangement realized by a pair (V,⊔li=1Bi).

2

• Fix a non-negative integer d. We say that a topological type is a topological typein class d if

(i) V is the disjoint union of k spheres and l is bounded by d(d− 1) + 2;(ii) V is the disjoint union of k spheres and a real projective plane, and l is

bounded by d(d−1)2 + 2.

Since k-sphere real del Pezzo surfaces of degree 1 or 2 have moduli, the classi-fication of topological types in class d on one surface, up to homeomorphism, maydiffer from that on another surface in the same deformation family. In this paper,a topological type S is said to be realizable in class d if there exist a k-sphere realdel Pezzo surface Xk of degree 1 (resp. 2) and a non-singular real algebraic curveA ⊂ Xk of class d such that the pair (RXk,RA) realizes S.

Let us present some general definitions and known results about real algebraiccurves. Let (X,σ) be any real algebraic non-singular compact curve. A very usefultool is Harnack-Klein’s inequality ([Har76], [Kle73]), which bounds the number l ofconnected components of RX by the genus g of X plus one. We say that (X,σ) is aM -curve or a maximal curve, if l = g + 1. If X \ RX is connected, we say that X isof type II or non-separating, otherwise of type I or separating ([Kle73]). Looking atthe real part of the curve and its position with respect to its complexification givesus information about l and viceversa. For example, we know that if X is maximal,then X is of type I. Or, if X is of type I then l has the parity of g+ 1. Moreover, if Xis of type I, the two halves of X \RX induce two opposite orientations on RX calledcomplex orientations of the curve ([Rok72]).

Harnack-Klein’s inequality, in combination with the adjunction formula, gives ob-structions on the number of connected components of any embedded real algebraiccurve. For any non-negative integer d, such obstruction has been taken into accountin Definition 1.3 when we define topological types in class d in the disjoint union of kspheres (and a real projective plane).

The anti-(bi)canonical system of a k-sphere real del Pezzo surface of degree 2,respectively degree 1 is a double cover of CP 2 ramified along non-singular real quartic,respectively a double cover of a quadratic cone in CP 3 ramified along a non-singularreal cubic section and the vertex. Conversely, any such double cover yields a delPezzo surface of degree 2, respectively 1. We mainly focus on the topological typeclassification of real algebraic curves in 4-sphere real del Pezzo surfaces of degree 2 andof degree 1. Later, we apply the classification tools developed for 4-sphere del Pezzosurfaces to k-sphere del Pezzo surfaces, with k < 4. In combination with variationsof classical classification methods, the main tools of construction of real curves relyon degeneration methods and those of obstruction of topological types rely on realenumerative geometry. The classifications in k-sphere real del Pezzo surfaces of degree2 are in Section 3 and the classifications in those of degree 1 in Section 4.

1.2 k-sphere real degree 2 del Pezzo surfacesLet us denote by Xk a k-sphere del Pezzo surface of degree 2, with 1 ≤ k ≤ 4.In Sections 3.1 - 3.3 we give definitions, notations and state the main results aboutXk. The proofs of the main statements are in Sections 3.4, 3.5, 3.6, 3.7 and 3.9.Moreover, Section 3.8 is devoted to the construction tools used in Section 3.9.

We say that a real algebraic curve in Xk is symmetric if it can be realized as liftingof a degree d real plane curve via the anti-canonical map of Xk. Moreover, we callsymmetric a topological type in class d in RXk if it is realizable by a symmetric realalgebraic curve of class d in Xk; Definition 3.6.We have a complete classification for class 1 and 2.

Result 1.4 (Proposition 3.8). For any topological type S in class d = 1, 2, there existsome Xk and a symmetric real algebraic curve of class d in Xk realizing S.

Fixed a topological type S in class d = 1, 2. One can construct a plane degreed real curve C and a plane real quartic Q̃ arranged in RP 2 so that S is realized in

3

Xk by a real curve A ⊂ Xk of class d, where Xk is the double cover of CP 2 ramifiedalong Q̃ and A is the lifting of C.Later, we focus on real algebraic curves of class d ≥ 3. First of all, Harnack-Klein’sinequality does not give a complete set of restrictions anymore. Besides an applicationof Comessatti-Petrovsky inequality ([Com28], [Pet33], [Pet38]) for real curves of evenclass d (Proposition 3.18), most of all classical obstructions do not seem to apply; forexample we did not find applications of the congruence results of [Rok72] and [GM77]to our setting.

In Section 3.4, in order to give new obstructions for topological types for everyinteger d, we use a variation of Bézout’s type restrictions exploiting Welschingerinvariants ([Wel05], [Shu15], [IKS15]), from real enumerative geometry.

For d = 3, dealing with real maximal curves only, we are able to obtain a partialclassification on 4-sphere real del Pezzo surfaces of degree 2.

Result 1.5 (Theorem 3.9). There are 74 topological types with 8 connected compo-nents in class 3 which are not prohibited by Bézout’s type restrictions. Moreover 48among the 74 topological types are such that: for each of them there exist some X4and a real maximal curve of class 3 in X4 realizing it. In addition 19 out of 48 arerealized by real symmetric curves.

There is one topological type in class 3 that we can realize by a non-singular realsymplectic curve (Proposition 3.11) in a real symplectic degree 2 del Pezzo surfacewith real part composed by 4 spheres. We do not know yet whether this topologicaltype is realizable algebraically.

Using the obstructions in Section 3.4 and combining the construction methodspreviously adopted to the case of X4, we obtain the following result on Xk withk < 4.

Result 1.6 (Proposition 3.10). There are respectively 79, 61 and 28 topological typeswith 8 connected components in class 3 which are not prohibited by Bézout’s typerestrictions. Moreover 49, 38 and 17 respectively among the 79, 61 and 28 topologicaltypes are such that: for each of them there exist some Xk and real maximal curve ofclass 3 in Xk realizing it, respectively with 1 ≤ k ≤ 3. In addition 6 out of respectively49, 38 and 2 out of 17 are realized by real symmetric curves.

Finally, we deal with non-symmetric topological types in class d on X4. We donot know yet if there are non-symmetric topological types realizable in class 3 and 4on X4. But, we have the following result.

Result 1.7 (Proposition 3.12). For any integer d ≥ 5, there exists a topological type Sin class d (consisting of 2d+ 1 connected components) such that there exists some X4and a non-symmetric real algebraic curve of class d in X4 realizing S. FurthermoreS is not realizable by any symmetric real algebraic curve of class d in any X4.

All symmetric topological types in class 3 are realized using construction tech-niques similar to those used for class 1 and 2 (Result 1.4). Such construction methodsdo not seem to be enough to realize all topological types in class d ≥ 3. Indeed,we use degeneration methods to realize some topological types in class 3 and some(non-symmetric) topological types in class d ≥ 5; see Proposition 3.33 and the end ofSection 3.9. Let us give a rough idea of the construction technique.

We start with a real degree 2 del Pezzo surface X0 with a real non-degeneratedouble point as only singularity and real part composed by k two-dimensional con-nected components, with 1 ≤ k ≤ 3. Then, we degenerate X0 to the union of areal ruled surface T and a k-sphere real del Pezzo surface S intersecting transverselyalong a curve E (Proposition 3.25). We construct real algebraic curves CT , CS ofgiven topology and homology class separately on T and on S; Section 3.8.1. More-over CT and CS have to intersect E in the same collection of points. Then, to end theconstruction, we use the version of patchworking developed by Shustin and Tyomkin([ST06a], [ST06b]) which allows us, under some transversality conditions, to "glue"such surfaces and curves to realize real algebraic curves on some Xk, respectively onsome Xk+1, with topology prescribed by the topological type realized by the triplet(RT ∪RS,RE,RCT ∪RCS); see Theorem 3.24. The patchworking technique presented

4

in [ST06a], [ST06b] has been recently exploited in [BDIM] to construct real algebraiccurves whose real part consists of a finite number of points in CP 2 and in the quadricellipsoid.

1.3 k-sphere real degree 1 del Pezzo surfacesLet us denote by Y k a k-sphere del Pezzo surfaces of degree 1, with 0 ≤ k ≤ 4.In Sections 4.1, 4.3, 4.4, 4.6 we give definitions, notations and state the main resultsabout k-sphere real del Pezzo surfaces of degree 1. The proofs are in Sections 4.2and 4.7. The obstruction presented in Proposition 4.2, called J -obstruction, gives acomplete set of restrictions for topological types up to class 3.

Result 1.8 (Proposition 4.13). For any topological type S in class d = 1, 2, 3, whichis not prohibited by the J -obstruction, there exist some Y k and a real algebraic curveof class d in Y k realizing S.

Moreover, in the case of Y 4, one can define a notion of positivity of the spheresinducing a refined classification (Section 4.4). Up to class 3, we show that the refinedand non-refined classifications are the same. Let us consider the disjoint union of areal projective plane and 4 spheres. Label two spheres as positive S2+, the others asnegative S2− and define

V + := S2+ t S2+ and V − := S2− t S2−.

Definition 1.9. Let⊔li=1Bi and

⊔li=1B

′i be two collection of l circles embedded in

V := RP 2 t V + t V −.

1. We say that the pairs (V,⊔li=1Bi) and (V,

⊔li=1B

′i) are refined homeomorphic

if they are homeomorphic via a homeomorphism f : V → V such that f(V ±) =V ±.

2. A topological type in V up to refined homeomorphism, is called a refined topo-logical type.

Result 1.10 (Theorem 4.12). For any refined topological type S in class d = 1, 2, 3,which is not prohibited by the J -obstruction, there exist some Y 4 and a real algebraiccurve of class d in Y 4 realizing S.

Furthermore, Proposition 4.11 gives Bézout-type restrictions for class d ≥ 4.

2 Preliminaries

2.1 Encoding topological types and real schemesLet X be a real algebraic surface equipped with a real structure σ : X → X.Let σ∗ : H2(X;Z) → H2(X;Z) be the group homomorphism induced by σ and letH−2 (X;Z) be the (−1)-eigenspace of σ∗. In the following, for a fixed homology classα ∈ H−2 (X;Z), we are interested in the classification of the topological types of thepair (RX,RA) up to homeomorphism, where A ⊂ X is a non-singular real algebraiccurve realizing α in H2(X;Z). The real part of A is homeomorphic to a union ofcircles embedded in RX, and can be embedded in RX in different ways. For thepurpose of this paper, we only need to explain how to encode the embedding of agiven collection

⊔i=1,..,lBi of l disjoint circles in RP 2 and in S2. An embedded circle

realizing the trivial-class in H1(RP 2;Z/2Z) or H1(S2;Z) is called oval, otherwise iscalled pseudo-line.

An oval in RP 2 separates two disjoint non-homeomorphic connected components:the connected component homeomorphic to a disk is called interior of the oval; theother one is called exterior of the oval. For each pair of ovals, if one is in the interiorof the other we speak about an injective pair, otherwise a non-injective pair.On the other hand, an oval in S2 bounds two disks; therefore, on S2 interior andexterior of an oval are not well defined. It follows that the encoding on S2 is not welldefined either and it depends on the choice of a point p on S2 \

⊔i=1,..,lBi. Let us

5

take S2 deprived of p, which is homeomorphic to R2 and let us call oval any circleembedded in R2. Analogously to the case of RP 2, in R2 one define interior and exte-rior of an oval and (non-)injective pairs for each pair of ovals.

We shall adopt the following notation to encode a given topological type realizedby a pair (V,

⊔i=1,..,lBi), where V is R2 or RP 2.

Notation 2.1. An empty union of ovals is denoted by 0. We say that a union of lovals realizes l if there are no injective pairs. The symbol 〈S〉 denotes the disjointunion of a non-empty collection of ovals realizing S, and an oval forming an injectivepair with each oval of the collection.We use the following notation only in Proposition 3.12, Proposition 3.32 and in theproof of the Proposition 3.12 in Section 3.9. Let h be a non-negative integer. Thesymbol N(h,S) denotes

• S, if h = 0;

• 〈N(h− 1,S)〉, otherwise.

Finally, the disjoint union of any two collections of ovals, realizing respectively S ′and S ′′ in V , is denoted by S ′ t S ′′ if none of the ovals of one collection forms aninjective pair with the ovals of the other one and they are both non-empty collections.A disjoint union of the form S ′t0 is still denoted S ′. Moreover, a pseudo-line in RP 2is denoted by J .

Definition 2.2.

• We say that the pair (S2,⊔i=1,..,lBi) realizes S if there exists a point p ∈

S2 \⊔i=1,..,lBi such that (S

2 \ {p},⊔i=1,..,lBi) realizes S.

• Let (RP 2,⊔i=1,..,lBi) and (V,

⊔i=1,..,l′ B

′i) be pairs respectively realizing S and

S ′, where V is a disjoint union of 2-spheres. We say that the pair (RP 2 tV,

⊔i=1,..,lBi t

⊔i=1,..,l′ B

′i) realizes S|S ′.

• Let (S2,⊔i=1,..,lBi) and (S

2,⊔i=1,..,l′ B

′i) be pairs respectively realizing S and

S ′. We say that the pair (S2 t S2,⊔i=1,..,lBi t

⊔i=1,..,l′ B

′i) realizes S : S ′.

Definition 2.3. Let (X,σ) be a real algebraic surface. A topological type S in RX,up to homeomorphism, is called real scheme. Let A ⊂ X be a real curve. We say thatA has real scheme S if the pair (RX,RA) realizes S, up to homeomorphism.

Finally, we need some more definitions for particular collections of ovals.

Definition 2.4. A collection of h ovals in RP 2 is called a nest of depth h if any twoovals of the collection form an injective pair. Let N1 and N2 be two nests of depth i1and i2 in RP 2. We say that the nests are disjoint if each pair of ovals, composed byan oval of N1 and an oval of N2, is non-injective.

Definition 2.5. A collectionNh of h ovals in S2 is a nest if each connected componentof S2 \Nh is either a disk or an annulus.Let Nik be k nests of depth ik in S2, with k ≥ 3. We say that the nests are disjointif a disk of S2 \Nij contains all other k − 1 nests, for all j ∈ {1, .., k}.

2.2 Hirzebruch surfacesA Hirzebruch surface is a compact complex surface which admits a holomorphic fibra-tion over CP 1 with fiber CP 1 ([Bea83]). Every Hirzebruch surface is biholomorphicto exactly one of the surfaces Σn = P(OCP 1(n)⊕C) for n ≥ 0. The surface Σn admitsa natural fibration

πn : Σn → CP 1

with fiber CP 1 =: Fn. Denote by Bn, resp. En, the section P(OCP 1(n)⊕ {0}), resp.P({0} ⊕ C). The self-intersection of Bn (resp. En and Fn) is n (resp. −n and 0).When n ≥ 1, the exceptional divisor En determines uniquely the Hirzebruch surface

6

since it is the only irreducible and reduced algebraic curve in Σn with negative self-intersection.

For example Σ0 = CP 1 × CP 1. The Hirzebruch surface Σ1 is the complexprojective plane blown-up at a point, and Σ2 is the quadratic cone with equationQ : X2 +Y 2−Z2 = 0 blown-up at the node in CP 3. The fibration of Σ2 (resp. of Σ1)is the extension of the projection from the blown-up point to a hyperplane section(resp. to a line) which does not pass through the blown-up point.

The group H2(Σn;Z) is isomorphic to Z⊕ Z and is generated by the classes [Bn]and [Fn]. An algebraic curve C in Σn is said to be of bidegree (a, b) if it realizesthe homology class a[Bn] + b[Fn] in H2(Σn;Z). Note that [En] = [Bn] − n[Fn] inH2(Σn;Z). An algebraic curve of bidegree (3, 0) on Σn is called a trigonal curve.

We can obtain Σn+1 from Σn via a birational transformation βpn : Σn 99K Σn+1which is the composition of a blow-up at a point p ∈ En ⊂ Σn and a blow-down ofthe strict transform of the fiber π−1n (πn(p)).

The surface Σn is also the projective toric surface which corresponds to the polygonof vertices (0, 0), (0, 1), (1, 1), (n+1, 0). The Newton polygon of an algebraic curve C ofbidegree (a, b) on Σn, lies inside the trapeze with vertices (0, 0), (0, a), (b, a), (an+b, 0).The surface Σn is canonically endowed by a real structure induced by the standardcomplex conjugation in (C∗)2. For this real structure RΣn is a torus if n is even anda Klein bottle if n is odd. We will depict RΣn as a quadrangle whose opposite sidesare identified in a suitable way. Moreover, the horizontal sides will represent REn.Furthermore, let C be any type I real algebraic curve in Σn, the depicted orientationon RC will denote a complex orientation of the curve.The restriction of πn to RΣn defines an S1-bundle over S1 that we denote by L. Weare interested in the isotopy types with respect to L of real algebraic curves in RΣn.

Definition 2.6.

• Let η be an arrangement of circles and points immersed in RΣn such that for anyimmersed point there exists a line of L intersecting the point with multiplicity 2and the multiplicity of intersection at each point of an immersed circle with thelines of L is at most 2. Such an arrangement, up to homeomorphism, is calledreal scheme. We say that a real algebraic curve C ∈ RΣn has real scheme η ifthe pair (RΣn,RC) realizes η, up to homeomorphism.

• Two real schemes in RΣn are L-isotopic if there exists an isotopy of RΣn whichbrings one arrangement to the other, each line of L to another line of L andwhose restriction to REn is an isotopy of REn.

• A real scheme in RΣn up to L-isotopy of RΣn is called an L-scheme.

• An L-scheme η is realizable by a real algebraic curve of bidegree (a, b) in Σn ifthere exists such a curve whose real part is L-isotopic to η.

• A trigonal L-scheme is an L-scheme in RΣn which intersects each fiber in 1 or3 real points counted with multiplicities and which does not intersect REn.

• A trigonal L-scheme η in RΣn is hyperbolic if it intersects each fiber in 3 realpoints counted with multiplicities.

2.3 Dessins d’enfantsOrevkov in [Ore03] has formulated the existence of real algebraic trigonal curvesrealizing a given trigonal L-scheme in RΣn in terms of the existence of a graph onCP 1. In the proof of Lemma 3.31 and of Proposition 4.20, we use this constructiontechnique.

Definition 2.7. Let n be a fixed positive integer. We say that a graph Γ is a realtrigonal graph of degree n if

7

• it is a finite oriented connected graph embedded in CP 1, invariant under thestandard complex conjugation of CP 1;

• it is decorated with the following additional structure:

– every edge of Γ is colored solid, bold or dotted;– every vertex of Γ is •, ◦, × (said essential vertices) or monochrome

and satisfying the following conditions:

1. P1R is a union of vertices and edges of Γ;2. any vertex is incident to an even number of edges; moreover, any ◦-vertex

(resp. •-vertex) to a multiple of 4 (resp. 6) number of edges;3. for each type of essential vertices, the total sum of edges incident to the

vertices of a same type is 12n;4. there are no monochrome cycles;5. the orientations of the edges of Γ form an orientation of ∂(CP 1 \ Γ) which

is compatible with an orientation of CP 1 \ Γ (see Fig. 3);6. all edges incidents to a monochrome vertex have the same color;7. ×-vertices are incident to incoming solid edges and outgoing dotted edges;8. ◦-vertices are incident to incoming dotted edges and outgoing bold edges;9. •-vertices are incident to incoming bold edges and outgoing solid edges.

Let n be a positive integer and let c(x, y) = y3+b2(x)y+b3(x) be a real polynomial,where bi(x) has degree in in x. By a suitable change of coordinates in Σn, any trigonalcurve C in Σn can be put into this form. Denote by ∆ = −4b32 +27b23 the discriminantof c(x, y) with respect to the variable y. The knowledge of the arrangement of the realroots of the real polynomials ∆ = −4b32+27b23, 27b23 and −4b32 in RΣn allows to recoverthe trigonal L-scheme realized by C in RΣn. Let f : CP 1 → CP 1 be the homogenizeddiscriminant, i.e. the rational function defined by f := ∆

27b23. Orevkov’s method allows

to construct real polynomials c(x, y) which have prescribed arrangements of the realroots and the construction is based on a consideration of the arrangement of thegraph given by f−1(RP 1) with the coloring and orientation induced by those of RP 1as depicted in Fig.1.1. In this section, we only give an algorithmic way to encode anytrigonal L-scheme in RΣn into a colored oriented graph on RP 1 ⊂ CP 1 just lookingat the intersections of the fibers of L with η; for details see [Bru07], [Deg12], [Ore03].

Fig. 1.1: Colored oriented RP 10∞ 1

Fig. 1.2: LI in RΣn

︷︸︸︷LI

Figure 1:

Definition 2.8. Let η be a trigonal L-scheme in RΣn. For any fixed intervalof points I := {(x, y) : y ∈ R, x ∈ [a, a + b] ⊂ R} ⊂ RΣn, we denote with LIthe fibers of L containing the points of I; see Fig. 1.2. Thanks to πn|RΣn we canencode η into a colored oriented graph Γ on RP 1 ⊂ CP 1 as follows (in Fig. 2the dashed lines denote fibers of L):

1. To each fiber of L intersecting η in two points, one of which with multi-plicity 2, we associate a ×-vertex on RP 1.

2. Fixed an interval I, let F1, F2 be two fibers of LI intersecting η in twopoints, one of which with multiplicity 2, such that η, up to L-isotopy, islocally as depicted in Fig. 2.2 or 2.3. Let F3 be another fiber betweenF1, F2. Then, we associate to F3 a ◦-vertex on RP 1. Moreover, if betweenF1 and F2 each other fiber intersects η in only one real point (as in Fig.2.2), then we associate to a fiber between F1 and F3 (resp. F3 and F2) a•-vertex on RP 1. Between • and ◦-vertices we put bold edges.

8

3. For all intervals I, except for the fibers of LI to which we associate essentialvertices and bold edges, we associate dotted (resp. solid) edges on RP 1 tothe fibers of LI which intersect η in three distinct real points (resp. onlyone real point).

4. The orientations of the edges incident to a vertex are in an alternatingorder. In particular, the orientations of the edges incident to an essentialvertex are respectively as described in 5− 7 of Definition 2.7.

The graph Γ, called real graph, is considered up to isotopy of RP 1, namely it isdetermined by the order of its colored vertices since the edges are determinedby the color of their adjacent vertices.We say that Γ is completable in degree n if there exists a complete real trigonalgraph Γ of degree n such that Γ ∩ RP 1 = Γ.

Fig. 2.1 Fig. 2.2 Fig. 2.3

F1 F3 F2

Fig. 2.4

F1F3F2

Fig. 2.5

Figure 2: Local topology of trigonal L-schemes and their corresponding real graphs.

Theorem 2.9 ([Ore03], [Deg12]). A trigonal L-scheme on RΣn is realizable by a realalgebraic trigonal curve if and only if its real graph is completable in degree n.

Given a real graph Γ, we depict only the completion to a real trigonal graph Γon a hemisphere of CP 1 since Γ is symmetric with respect to the standard complexconjugation. Moreover, we can omit orientations in figures representing real trigonalgraphs because each vertex is adjacent to an even number of edges oriented in analternating order as, for example, depicted in Fig. 3 and such orientations are com-patible with each others.

••

×

Figure 3: Colored vertices of a real trigonal graph.

3 Real curves on k-sphere real del Pezzo surfaces ofdegree 2

3.1 DefinitionsLet X be CP 2 blown up at seven points in generic position; then, the surface X isa del Pezzo surface of degree 2 (see [Rus02], [Dol12, Chapter 8]). The anti-canonicalsystem φ of X is a double ramified cover of CP 2 and the branch locus of φ consists ofan irreducible non-singular quartic Q defined by a homogeneous polynomial f(x, y, z).By construction, the anti-canonical class c1(X) is the pull back via φ of the class of aline in CP 2 ([DIK00]). Moreover, the surface X is isomorphic to the real hypersurfacein CP (1, 1, 1, 2) defined by the weighted polynomial equation f(x, y, z) = w2, withcoordinates x, y, z and w respectively of weights 1 and 2. Conversely, any doublecover of CP 2 ramified along a non-singular algebraic quartic yields a del Pezzo surfaceof degree 2.

9

If one equips X with a real structure σ, the quartic Q is real and f(x, y, z) canbe chosen with real coefficients and so that the real surface (X,σ) is isomorphic tothe real hypersurface in CP (1, 1, 1, 2) of equation f(x, y, z) = w2. It follows that thedouble cover φ projects RX into the region

Π+ := {[x : y : z] ∈ RP 2 : f(x, y, z) ≥ 0}.

Conversely, the double cover of CP 2 ramified along a non-singular real quartic Q ⊂CP 2 and a choice of a real polynomial equation f(x, y, z) of Q yields a real del Pezzosurface X.

φ

Figure 4: Example: φ : RX 7→ Π+.

The surface X is a R-minimal1 if and only if X is a 4-sphere or a 3-sphere real delPezzo surface (Definition 1.1). Moreover X is a k-sphere real del Pezzo surface, with1 ≤ k ≤ 4 if and only if Q is a non-singular real quartic realizing the real scheme kin RP 2 and Π+ is orientable; see [DK02] and, as example for k = 4, look at Fig. 4,where Π+ is in gray on the right.

Notation 3.1. Let X be a k-sphere real del Pezzo surface of degree 2. We denotethe connected components of RX with X1, . . . , Xk.

The lifting of a non-singular real algebraic curve C ⊂ CP 2 of degree d via φ is areal algebraic curve A of class d in X (Definition 1.2). Moreover, from the topologicaltype realized by the triplet (RP 2,RQ,RC), one recovers the real scheme of the pair(RX,RA). As example, assume that d = 1, the quartic Q is maximal, Π+ is orientableand RQ ∪ RC is arranged in RP 2 as depicted in Fig. 4 on the right; then the pair(RX,RA) has real scheme 1 : 1 : 0 : 0; Fig. 4 on the left.

If X is a 4-sphere real del Pezzo surface of degree 2, any real algebraic curve hasclass d, where d is some non-negative integer ([Rus02]).

Combining Harnack-Klein’s inequality and the adjunction formula, one obtainsthe following immediate result.

Proposition 3.2. Let A be a real algebraic curve of class d in a k-sphere real delPezzo surface X of degree 2, with 1 ≤ k ≤ 4. Then, the number l of ovals of RA isbounded as follows:

l ≤ d(d− 1) + 2.

3.2 Real schemesDefinition 3.3. Let X be a k-sphere real del Pezzo surface X of degree 2, with0 ≤ k ≤ 4. Let us denote SDP2(X, k) the set of all real schemes in X.Notice that SDP2(X, k) does not depend on the choice of X. Therefore, from now on,we omit X and write SDP2(k).

Let us enrich the notion of real scheme with some extra conditions deriving fromProposition 3.2.

Definition 3.4. Let S be in SDP2(k). We say that S is in class d or we writeS ∈ SDP2(k, d) if the number of ovals of S does not exceed d(d− 1) + 2.

Definition 3.5. We say that S ∈ SDP2(k, d) is realizable in Xk and in class d, ifthere exist a k-sphere real del Pezzo surface Xk of degree 2 and a real algebraic curveA ⊂ Xk of class d, such that the pair (RXk,RA) realizes S.

1We say that a real algebraic variety (X,σ) is R-minimal, if every real degree 1 holomorphicfunction f : X → Y to a real algebraic surface (Y, τ) is a biholomorphism.

10

To refine the classifications of real schemes in SDP2(k, d), one can additionally askabout the realizability of a given real scheme by symmetric real curves.

Definition 3.6.

• Let A be a class d algebraic curve in a k-sphere real degree 2 del Pezzo surfaceX, with 1 ≤ k ≤ 4. Let φ : X → CP 2 be the anti-canonical map of X. We saythat A is symmetric if it is the lifting via φ of a plane algebraic curve of degreed. Otherwise, we say that A is non-symmetric.

• We say that S ∈ SDP2(k, d) is symmetric in class d if it is realizable in Xk andin class d by a symmetric real algebraic curve of class d. Otherwise, we say thatS is non-symmetric in class d.

Let us lighten the real scheme notation introduced in Section 2.1.

Notation 3.7. Let S := S1 : . . . : S4 be a topological type in the disjoint union of 4spheres. Let Xk be a k-sphere real del Pezzo surface of degree 2, with 1 ≤ k ≤ 4. IfS has at least 4− k trivial entries, we say that S is a real scheme in RXk.

3.3 Main resultsThe real scheme classification is complete in class 1 and 2. The proof of the followingstatement is in Section 3.5.

Proposition 3.8 (Class 1 and 2). Any real scheme S ∈ SDP2(k, d) where d = 1, 2and 1 ≤ k ≤ 4, is realizable in Xk and in class d. Moreover S is symmetric in classd.

For real schemes in class d ≥ 3, Proposition 3.2 does not provide a completesystem of restrictions anymore. In Section 3.4, we show how to use Welschinger-typeinvariants to obtain Bézout-type obstructions for any class d and, for d even, wepresent an application of Comessatti-Petrovsky inequality.

Sections 3.5 - 3.9 are devoted to construction of real algebraic curves; severalconstruction techniques are combined, including dessins d’enfants and recent devel-opments of Viros’s patchworking method.

We mainly focus on classifications on 4-sphere del Pezzo surfaces of degree 2: wegive a partial classification of real schemes in class 3 with 8 ovals (Theorem 3.9), andwe realize some non-symmetric real schemes for each class d ≥ 5 with 2d + 1 ovals(Proposition 3.12). In addition, we use the classification tools adopted on 4-spheredel Pezzo surfaces of degree 2 in the case of k-sphere real degree 2 del Pezzo surfaces,with k < 4 (Proposition 3.10).

Theorem 3.9 (k = 4 and Class 3). There are 74 real schemes in SDP2(4, 3) with 8ovals, which are non-prohibited by Proposition 3.14 and Lemmas 3.16, 3.17 . Amongthose 48 are realizable in X4 and in class 3. Moreover 19 out the 48 are symmetricin class 3.

Proposition 3.10 (k = 3, 2, 1 and Class 3). There are respectively 79, 61 and 28real schemes in SDP2(k, 3) with 8 ovals, which are non-prohibited by Proposition 3.14for k = 3, 2 and Lemma 3.15 for k = 1. Among those respectively 49, 38 and 17are realizable in Xk and in class 3. Moreover 6 and 2 are symmetric in class 3,respectively for k = 3, 2 and k = 1.

The realization of the real schemes in SDP2(k, 3) of Theorem 3.9 and Proposition3.10 are in the proof of Proposition 3.21 (symmetric ones) and of Proposition 3.33.On the last page of the paper, it is presented:

• a summary of the realized (symmetric) real schemes in Xk and in class 3, with1 ≤ k ≤ 4 (Table 3).

• a list of real schemes in class 3 which are still unrealized on Xk, with 1 ≤ k ≤ 3;but, which can not be realized by class 3 real curves on 4-sphere real del Pezzosurface of degree 2 (Table 4).

11

The real scheme 2 t 〈1〉 t 〈1〉 t 〈1〉 : 0 : 0 : 0 is symplectically realizable in X4 andin class 3 (proof in Section 3.7), but we do not know yet whether it is realizablealgebraically.

Proposition 3.11. There exist a 4-sphere real symplectic degree 2 del Pezzo surfaceX and a non-singular real symplectic curve of class 3 in X realizing the real scheme2 t 〈1〉 t 〈1〉 t 〈1〉 : 0 : 0 : 0.

For each class d ≥ 5, we realize some (non-symmetric) real schemes in class d.The proof of the following statement is at the end of Section 3.9.

Table 1: (Non-symmetric) real schemes S1 : S2 : N(h3, 0) : N(h4, 0) realized inX4 and in class d ≥ 5

S1 : S2 Extra conditionsk1 = 0, h2 6= 1

(1) 1 tN(h1, 1) : 1 tN(k2, 1 t 〈1〉) tN(h2, 0) h1 ≡ 1 mod 2If h2 = 2, k2 ≡ 1 mod 2.If h2 = 0, k2 6= 0, 1.k1 = 0, h1 > 1

(2) 1 t 〈1〉 tN(h1, 1) : 1 tN(k2, 1) tN(h2, 0) h2 6= 1, h2 6= k2 + 1If h2 = 0, k2 ≡ 1 mod 2.h1 6= 1, k1 6= 0h1 6= k1 + 1

(3) 1 tN(k1, 1) tN(h1, 0) : N(k2, 1 t 〈1〉) tN(h2, 1) If h1 = 0, k1 ≡ 1 mod 2.If h2 = 0, 1, k2 6= 0.If h2 = 0, k2 6= 1.h2 6= 1, k2 6= 0

(4) 1 tN(k2, 1 t 〈1〉) tN(h2, 0) : N(k1, 1) tN(h1, 1) h1 + k1 ≡ 1 mod 2If h2 = 0, 2, k2 6= 1.If h2 = 2, k2 ≡ 1 mod 2.k1 = 0, h1 6= 1

(5) 1 tN(h1, 0) tN(h2, 1) : N(k2 + 1, 1 t 〈1〉) h2 6= 0, h1 6= h2 + 1k2 6= 1 ( =⇒ d ≥ 6)k1 = 0, h1 6= 1h2 6= 0 ,h1 6= h2 + 1

(6) 1 t 〈1〉 tN(h1, 0) tN(h2, 1) : N(k2 + 1, 1) k2 ≡ 1 mod 2h1 6= 2h1 6= 1h1 6= h2 + 1

(7) N(k2, 1 t 〈1〉) tN(h2, 1) tN(h1, 0) : N(k1 + 1, 1) k1 ≡ 1 mod 2If (h1, h2) = (1, 1), (2, 0),k2 ≡ 1 mod 2.h1 6= h2 + 1, k2 6= 1, 2

(8) N(k1 + 1, 1) tN(h2, 1) tN(h1, 0) : N(k2, 1 t 〈1〉) h1 6= k1 + 2h2 6= k1 + 1

(i) Central column: the symbol N(h3, 0) : N(h4, 0) is omitted because common to everyreal scheme.(ii) Right column: sufficient conditions on the parameters ki, hj , d to have non-symmetric real schemes.

Proposition 3.12 (Non-symmetric and Class d ≥ 5). Let d, k1, k2, and h1, h2, h3, h4be non-negative integers such that

• d ≥ 5;

• k1 + k2 = d− 4;

•4∑i=1

hi = d− 1;

• h3 ≡ 1 mod 2, if h3 6= 0;

12

• h4 ≡ 1 mod 2, if h4 6= 0.

Then each real scheme S1 : S2 : N(h3, 0) : N(h4, 0) in SDP2(4, d) in Table 1, isrealizable in X4 and in class d. Moreover, the real schemes whose parameters respectthe extra conditions in Table 1, are non-symmetric in class d.

3.4 Obstructions and Welschinger-type invariantsWelschinger invariants can be regarded as real analogues of genus zero Gromov-Witteninvariants. They were introduced in [Wel05] and count, with appropriate signs, thereal rational curves which pass through a given real collection of points in a given realrational algebraic surface. In the case of k-sphere real del Pezzo surfaces of degree 2,the Welschinger invariants, as well as their generalizations to higher genus ([Shu15]),can be used to prove the existence of interpolating real curves of genus 0 ≤ g ≤ k− 1;see [IKS15] and [Shu15].

Proposition 3.13. [Shu15, Propositions 4 and 5] Let s be an integer greater than 1and r1, r2 be two non-negative odd integers such that r1 +r2 = 2s. Let P be a genericconfiguration of 2s+j real points, with j = 2, 1, 0, on a k-sphere real del Pezzo surfaceX of degree 2, where k = 2 + j, such that

• Xi contains ri points of P, with i = 1, 2;

• Xi contains one point of P, if i 6= 1, 2.

Then, there exists a real algebraic curve T of class s and genus j + 1 in X passingthrough P. Furthermore, the points of P belong to the one-dimensional connectedcomponents of RT .

We use the result of Proposition 3.13 to prove the following proposition.

Proposition 3.14. Let s be an integer strictly greater than 1 and r1, r2 be two non-negative odd integers such that r1 + r2 = 2s. Moreover, let A be a non-singular realalgebraic curve of class d in a k-sphere real del Pezzo surface X of degree 2, withk = 4, 3, 2. Let t denote the number of connected components of RX which intersectRA. Assume that RA has ri disjoint nests Nh of depth jh on Xi, respectively with1 ≤ h ≤ r1 for i = 1, and with r1 + 1 ≤ h ≤ 2s for i = 2.

(1) If r1, r2 > 1, then2s∑h=1

jh ≤ ds− (t− 2);

(2) If r1 = 2s− 1 and r2 = 1, then2s−1∑h=1

jh ≤ ds− (t− 1).

Proof. Assume that r1 and r2 are strictly greater than 1. It follows that RA has atleast 3 disjoint nests on Xi, with i = 1, 2. In order to prove inequality (1), let uschoose a generic collection P of 2s + j real points, with j = k − 2, in the followingway. On each boundary of the r1 (resp. r2) disks in X1\

r1th=1

Nh (resp. X2\2st

h=r1+1Nh),

pick a point. Moreover, pick a point on every connected component Xi, with i = 3, 4,such that the point belongs to RA any time the real algebraic curve has at least oneoval on Xi. Then, Proposition 3.13 assures the existence of a real algebraic curveT of class s and genus j + 1 on X passing through P. Furthermore, the points ofP belong to the one-dimensional connected components of RT . Thus, the number

of real intersection points of A with T is at least 2(2s∑h=1

jh + (t − 2)). Inequality (1)

follows directly from the fact that the intersection number A ◦ T = 2ds is greater orequal than the number of real intersection points of A with T .The proof of (2) is similar to the previous one.

The following statement gives more topological obstructions for real curves in1-sphere real del Pezzo surfaces of degree 2.

Lemma 3.15. Let A be a real algebraic curve in class d in a 1-sphere real del Pezzosurface X of degree 2. Assume that RA has three nests Nh of depth jh on X1. Then,

j1 + j2 + j3 ≤ 2d.

13

Proof. The statement follows from

• the existence of a real algebraic curve T of class 2 and genus 0 on X passingthrough a given real configuration of 3 distinct points on X1 (see [IKS15, Table1, Section 2.2]) and

• an argument similar to that used in the proof of Proposition 3.14.

We have one more Bézout-type restriction on the topology of real curves in 4-sphere real del Pezzo surfaces of degree 2.

Lemma 3.16. Let A be a real algebraic curve in class d in a 4-sphere real del Pezzosurface X of degree 2. Assume that RA has a nest N1 of depth j1 on X1 and a nestN2 of depth j2 on X2. Let t denote the number of connected components of RX whichintersect RA. Then,

j1 + j2 ≤ 2d− (t− 2).

Proof. Let |L| be a linear system of curves on X. Passing through a given (real) pointon X defines a (real) hyperplane on |L|. Therefore, given a (real) configuration of hpoints on X such that h is less or equal to the dimension of |L|, there exists a (real)curve in |L| passing through the h points. It follows that there exists a real algebraiccurve T of class 2 and genus 3 on X passing through a given real configuration P of6 distinct points such that Xi contains 2 points of P, with i = 1, 2, and Xi containsone point of P, if i 6= 1, 2. The existence of such a curve T and an argument similarto that used in the proof of Proposition 3.14, prove the statement.

A variant of the technique used in proof of Lemma 3.16 leads to prohibit a par-ticular real scheme in class 3 in 4-sphere real del Pezzo surfaces of degree 2.

Lemma 3.17. There is no real algebraic curve of class 3 in any 4-sphere real delPezzo surface X of degree 2 realizing the real scheme

S := 〈1〉 t 〈1〉 t 〈1〉 t 〈1〉 : 0 : 0 : 0.

Proof. Assume that there exists a non-singular real algebraic curve A of class 3 real-izing S in X. Let us choose a configuration P of 6 real points as follows. On eachboundary of the 4 disks in X1 \ RA, pick a point. Moreover, pick a point on theconnected components X2 and X3. Then, there exists a non-singular real algebraiccurve T of class 2 passing through P and T has at most two ovals on X1 and one ovalon both X2 and X3. Thus, the number of real intersection points of A with T is atleast 14. But the intersection number A ◦ T is 12.

In this article, apart form Proposition 3.12, we mainly focus on class 3 real curveson Xk. To look further, in Proposition 3.18 we present one possible application ofComessatti-Petrovsky inequality ([Com28], [Pet33], [Pet38]), which gives a topologicaltype restriction for real curves of even class d.

Proposition 3.18. Let A be a non-singular real algebraic maximal curve of evenclass d ≥ 8 in a k-sphere real del Pezzo surface X of degree 2, with 1 ≤ k ≤ 4. Then,the pair (RX,RA) does not realize the real scheme l : 0 : 0 : 0, with l = d(d− 1) + 2.

Proof. Let Y π→ X be a double cover of X ramified along A. Let B1 and B2 be twodistinct disjoint unions of connected components of RX \ RA such that each Bi isbounded by RA. There exist two lifts σ1, σ2 to Y of the real structure of X via thedouble cover π and the real part of Y is the double of one of the Bi’s. Thanks toComessatti-Petrovsky inequality one has

− 14− d(3d− 2)2

≤ χ(RY ) ≤ 16 + d(3d− 2)2

, (1)

where χ(RY ) = 2χ(Bi).For each possible choice of B1 and B2 (Remark 3.19), a direct application of (1) showsthat at least one χ(Bi) does not respect the inequalities.

14

In order to find other Comessatti-Petrovsky restrictions for real schemes in classd, the following should be observed.

Remark 3.19. In the proof of Proposition 3.18, the choices of B1 and B2 are notindependent; in fact, the choice on Xi of two disjoint unions of connected componentsof Xi \ RA bounded by RA imposes the choice on the other spheres. However, it isnot obvious to know which are these imposed halves, unless you already know thedouble covering Y which contains in particular the information on B1 and B2.

3.5 Class 1 and 2Let us construct some symmetric real curves of class 1 and 2 on k-sphere real delPezzo surfaces of degree 2.

Proof of Proposition 3.8. Fix a non-negative integer k ≤ 4 and any real scheme S inSDP2(k, d), with d = 1, 2. It is easy to construct a non-singular real plane quartic Q,with real scheme k, and a line (resp. conic) C in CP 2 such that, via the double coverof CP 2 ramified along Q, the lifting of C is a class 1 (resp. 2) real algebraic curverealizing S in a k-sphere real del Pezzo surface of degree 2. See Example 3.20.

Example 3.20. Let us fix k = 4. From the quartics and lines arranged in RP 2 as inFig. 5, one construct 4-sphere real del Pezzo surfaces X of degree 2 and class 1 realalgebraic curves in X realizing all real schemes in class 1.Analogously, to realize all real schemes in X4 and class 2 with a maximal numberof ovals, it is enough to construct real plane conics and quartics mutually arrangedin RP 2 as in Fig. 6. The construction of such a pair of real plane curves, realizingthe first three real schemes of Fig. 6, follows from Hilbert’s construction method([Hil02]) which allows to construct a real quartic perturbing the union of two realconics; the remaining arrangements in Fig. 6 are realized fixing a real quartic whichhas a non-convex oval, taking a pair of lines which intersect the quartic transversally,and perturbing the lines to a conic; see [Bru21] for details on perturbations.

Figure 5: Arrangements of real lines (in thick black) and real maximal quartics (ingray) in RP 2.

Figure 6: Arrangements of real conics (in thick black) and maximal quartics (in gray)in RP 2.

3.6 Class 3Let us prove a part of Theorem 3.9 and Proposition 3.10, realizing some symmetricreal schemes in Xk and in class 3.

15

Figure 7: Mutual arrangements, up to isotopy, on RP 2 of a real maximal cubic (inthick black) and a real maximal quartic (in gray).

Fig. 8.1 Fig. 8.2 Fig. 8.3

Figure 8: 1− 2: Intermediate constructions on RP 2. 3: Mutual arrangement on RP 2of a real maximal cubic (in thick black) and a real maximal quartic (in gray).

Fig. 9.1 Fig. 9.2 Fig. 9.3 Fig. 9.4 Fig. 9.5 Fig. 9.6

Fig. 9.i Fig. 9.ii Fig. 9.iii Fig. 9.iv Fig. 9.v Fig. 9.vi

Figure 9: 1− 6: Intermediate constructions on RP 2. i− vi: Mutual arrangements onRP 2 of real maximal cubics (in thick black) and real quartics Q (in gray).

Proposition 3.21. Each real scheme S in SDP2(k, 3) labeled with ◦ and/or ◦∗ inTable 3, is symmetric in class 3.

Proof. In [Ore02], Orevkov has constructed real maximal quartics Q and cubics Carranged, up to isotopy, in RP 2 as depicted in Fig. 7. To all such pairs correspondreal algebraic curves of class 3 in 4-sphere real del Pezzo surfaces of degree 2 realizing12 real schemes among those labeled with ◦ in Table 3.Let us realize the real scheme 2 : 2 : 2 : 2. There exist a real cubic C̃ and a real line Lin CP 2 arranged in RP 2 as represented in Fig. 8.1. Let p̃(x, y, z) = 0 and l(x, y, z) = 0be real polynomial equations respectively defining C̃ and L. Pick three real lines L1,L2, L3, as those depicted in dashed in Fig. 8.1. Take a small perturbation of C̃replacing p̃(x, y, z) with p(x, y, z) := p̃(x, y, z) + εl1(x, y, z)l2(x, y, z)l3(x, y, z), whereli(x, y, z) is a real polynomial defining Li, with i = 1, 2, 3, and ε 6= 0 is a sufficientsmall real number. Up to a choice of the sign of ε, the real curve C, defined byp(x, y, z) = 0, is a real cubic arranged in RP 2 as depicted (in thick black) in Fig.

16

8.2. Let⋃4i=1 Li be the union of four non-real lines pairwise complex conjugated and

defined by a real polynomial u(x, y, z). Take a small perturbation of C̃ ∪ L replacingp̃(x, y, z)l(x, y, z) with p̃(x, y, z)l(x, y, z) + δu(x, y, z) = 0, where δ 6= 0 is a sufficientsmall real number. Up to the choice of the sign of δ, such equation defines a non-singular real plane maximal quartic Q such that Q∪C is arranged in RP 2 as picturedin Fig. 8.3. It follows that 2 : 2 : 2 : 2 is realizable in X4 and in class 3.Now, we end the proof realizing the real schemes in SDP2(k, 3) listed below. Thereexist a real quartic Q with real scheme k, where 2 ≤ k ≤ 4 (resp. 1 ≤ k ≤ 4), andthree real lines arranged in RP 2 as pictured in Fig. 9.1 - 9.4 (resp. Fig. 9.5 and 9.6);where we depict only the ovals of RQ (in gray) intersecting the three lines (in thickblack). Perturb the union of the three lines into a non-singular real cubic C such thatC ∪ Q is arranged in RP 2 respectively as depicted in Fig. 9.i - 9.vi; we depict onlythe ovals of RQ intersecting the cubic (in thick black). From C ∪Q, one realizes thefollowing real schemes in Xk and in class 3:

• for 2 ≤ k ≤ 4,

2 t 〈4〉 : 1 : 0 : 0, 6 : 2 : 0 : 0, 1 t 〈4〉 : 2 : 0 : 0, 3 t 〈1〉 t 〈1〉 : 1 : 0 : 0

• for 1 ≤ k ≤ 4,1 t 〈6〉 : 0 : 0 : 0, 〈1〉 t 〈5〉 : 0 : 0 : 0.

3.7 Symplectic curve on a 4-sphere real symplectic degree 2del Pezzo surface

There exists a certain mutual arrangement in RP 2 of a real symplectic cubic and a realsymplectic quartic which is unrealizable algebraically; see [Ore02]. Analogously to thealgebraic case, one can construct from such arrangement in RP 2 a real symplectic delPezzo surface of degree 2 and a real symplectic curve of class 3 on it with topologyprescribed from the arrangement on the real projective plane.

Proof of Proposition 3.11. Let us consider (CP 2, ωstd, conj), where ωstd is the sym-plectic Fubini-Study 2-form on CP 2 and conj : CP 2 → CP 2 is the standard realstructure on CP 2. Let conj∗ : H2(CP 2;Z) → H2(CP 2;Z) be the group homomor-phism map induced by conj. It follows that conj∗ωstd = −ωstd. Due to [Ore02],there exist a non-singular real symplectic maximal quartic Q and a non-singular realsymplectic maximal cubic C which are mutually arranged in RP 2 as depicted in Fig.10. The double cover φ : X → CP 2 ramified along Q carries a natural symplec-tic structure ω such that ω = φ

∗ωstd ([Gro13],[Aur00]). Let σ be one of the two

lifts of conj via the double ramified cover. Since φ ◦ σ = conj ◦ φ, we have thatσ∗ω = −ω; namely σ : X → X is a real structure of X. Then, up to choose σ,the surface (X,ω, σ) is real diffeomorphich to a 4-sphere real del Pezzo surface ofdegree 2 and, from C, we construct a real symplectic curve of class 3 in X realizing2 t 〈1〉 t 〈1〉 t 〈1〉 : 0 : 0 : 0.

Figure 10:

3.8 Constructions via patchworkingIn this section, we present a construction method that allows to construct (non-symmetric) real algebraic curves with prescribed topology in k-sphere real del Pezzo

17

surfaces of degree 2. First of all, let us give some definitions.Let Blp1,..,p7 : S → CP 2 be the blow-up of CP 2 at a collection of 7 points p1, .., p7subject to the condition that all of them do not belong to a conic, 6 of them belongto a conic, no 3 of them belong to a line. Then, the strict transform of the conicpassing through 6 points of the collection is a smooth rational curve ES ⊂ S of self-intersection (−2) in S.Assume from now on, that S contains a unique smooth rational curve of self intersec-tion (−2). The pair (S,ES) is called a 1-nodal degree 2 del Pezzo pair.The anti-canonical system φ′ of S decomposes into a regular map S → S′ of degree 1which contracts the (−2)-curve of S, and a double cover S′ → CP 2 ramified along aquartic Q̃ with a double point as only singularity. Let us call the surface S′ a 1-nodaldel Pezzo surface of degree 2. Conversely, the minimal resolution of the double coverof CP 2 ramified along a quartic with a double point as only singularity is a 1-nodaldegree 2 del Pezzo pair.

Let us equip S with a real structure σ′, then ES is real. Assume that RS ishomeomorphic to

⊔ji=1 S

2. It follows that the quartic Q̃ ⊂ CP 2 is real, it has a realnon-degenerate double point as only singularity and that RQ̃ consists of j connectedcomponents of dimension 1. Conversely, given such a quartic, one can construct a1-nodal degree 2 del Pezzo pair (S,ES) where S is equipped with a real structure suchthat RS is homeomorphic to

⊔ji=1 S

2; see [DIK00]. The homology group H−2 (S;Z) isgenerated by c1(S) and ES ([Rus02]).Let us give some more definitions.

Definition 3.22.

• Let (S,ES) be a 1-nodal degree 2 del Pezzo pair. Let S be equipped with areal structure σ′. If RS is homeomorphic to

⊔ji=1 S

2, we say that (S,ES) is aj-sphere real 1-nodal degree 2 del Pezzo pair.

• Let A ⊂ S be a real algebraic curve realizing the class dc1(S)+k[ES ] ∈ H2(S;Z).Then, we say that A has bi-class (d, k).

Notation 3.23. Let s be a non-negative integer greater or equal to 2. We denotewith

∨sj=1 S

n a bouquet of s n-dimensional spheres.

Topological construction: Let X ′0 be a real reducible surface given by the unionof two real algebraic surfaces S and T , where

(1) T is a non-singular real quadric surface;

(2) S contains a unique smooth rational (−2)-curve ES ⊂ S such that (S,ES) is aj-sphere real 1-nodal degree 2 del Pezzo pair, with 1 ≤ j ≤ 3;

(3) S and T intersect transversely along a real curve E which is a bidegree (1, 1)real curve in T and ES in S.

Let CS ⊂ S and CT ⊂ T be non-singular real algebraic curves respectively of bi-class(d, k) and of class kE in H2(T ;Z). Both CS and CT intersect transversely E in thesame real configuration of 2k distinct points; i.e.

E ∩ CS = E ∩ CT .

If T is the quadric ellipsoid (resp. T is the real quadric surface with empty realpart) and RE = ∅, the topological type S realized by (RS ∪ RT,RCS ∪ RCT ) is anarrangement of ovals in

⊔j+1j=1 S

2 (resp.⊔jj=1 S

2).

Otherwise, if T is the quadric ellipsoid (resp. T is the quadric hyperboloid) andRE ' S1, from the topological type realized by the pair (RS ∪ RT,RCS ∪ RCT ), wecan realize an arrangement S of ovals in

⊔j+1j=1 S

2 (resp.⊔jj=1 S

2) as follows.Locally RT ∩RS is given as the intersection of two real planes as depicted in Fig. 11.1,and (RS ∪ RT ) \ RE has 4 connected components W1, W2 ⊂ RS and H1, H2 ⊂ RT(Fig. 11.2). We can glue W1 either to H1 or to H2 along RE. After making a choice

18

for W1, we glue W2 to the remaining connected component along RE (Fig. 11.3).Either choices of gluing the four connected components give us the disjoint union ofj + 1 (resp. j) spheres, and from RCS ∪ RCT we get an arrangement S of ovals in⊔j+1j=1 S

2 (resp.⊔jj=1 S

2). Example in Fig. 11.4 and 11.5.

Fig. 11.1 Fig. 11.2 Fig. 11.3 Fig. 11.4 Fig. 11.5

Figure 11:

By Theorem 3.24, such topological construction is realizable algebraically.The proof of Theorem 3.24 requires the existence of a real flat one-parameter familywhose general fibers are (j + 1)-sphere, respectively j-sphere real del Pezzo surfacesof degree 2 and whose central fiber is X ′0. We prove the existence of such a family inCorollary 3.26.

Theorem 3.24. The real scheme S is realizable in Xk and in class d, with k = j,respectively k = j + 1.

Proof. Due to Corollary 3.26, we can put X ′0 in a real flat one-parameter familyπ̃′ : X′ → D(0), where X′ is a 3-dimensional real algebraic variety and D(0) ⊂ Cis a real disk centered at 0, such that the fibers X ′t =: π̃′−1(t) are (k-sphere real)non-singular degree 2 del Pezzo surfaces (with k = j, resp. k = j + 1), for t 6= 0(and t real), and the central fiber is X ′0. By Ramanujan’s Vanishing theorem ([Dol12,Section 8])

H1(X ′0;OX′0(C0)) = 0;then, [ST06a, Theorem 2.8] assures the existence of an open neighborhood U(0) ⊂D(0) and a deformation Ct in π̃′−1(t) such that Ct are non-singular (real) curves inX ′t for t (real) in U(0) \ {0}. Moreover, there exists a real t̃ ∈ U(0) \ {0} such thatthe pair (RX ′

t̃,RCt̃) realizes the real scheme S.

To prove Corollary 3.26, we need the following proposition. In the proof we makeuse of a type of construction presented in [Ati58], which found recent applications inreal enumerative geometry; see [BP13]. Similar constructions can be found in [MT88],[MT90] and [MT94].

Proposition 3.25. Let Q̃ be a real quartic in CP 2 with a real non-degenerate doublepoint q as only singularity and such that RQ is homeomorphic to either

⊔ji=1 S

1t{q}or to

⊔j−1i=1 S

1t∨2j=1 S

1, where 1 ≤ j ≤ 3. Then, there exists a real flat one-parameterfamily of non-singular (k-sphere real) degree 2 del Pezzo surfaces (with k = j, resp.with k = j + 1) but the central fiber which is a real reducible surface X ′0 equal to theunion of two real algebraic surfaces S ∪ T , where

(1) T is either a real quadric hyperboloid or a real quadric surface with empty realpart (resp. a real quadric ellipsoid);

(2) S is the minimal resolution of the double cover of CP 2 ramified along Q̃ and itcontains a unique smooth rational curve ES ⊂ S such that (S,ES) is a j-sphere1-nodal degree 2 del Pezzo pair, with 1 ≤ j ≤ 3;

(3) S and T intersect transversely along a curve E which is a bidegree (1, 1) realcurve in T and the (−2)-curve ES in S.

Proof. Let f(x, y, z) = 0 be a real polynomial equation defining the real quartic Q̃ inCP 2. Up to multiply f(x, y, z) by −1, we can always put Q̃ in a real flat one-parameterfamily π : Q→ D(0), where

19

• D(0) ⊂ C equipped with the standard real structure of C, is a real disk centeredat 0;

• Q ⊂ CP 2 ×D(0) is defined by f(x, y, z) + εz4t2 = 0, with ε ∈ {1,−1}

and such that

• the fibers Qt := π−1(t) are non-singular (real) quartics (with real part homeo-morphic to either

⊔ji=1 S

1 or to⊔j+1i=1 S

1, and Π+ is orientable) for t 6= 0 (andt real);

• Q0 = Q̃.

From the family of quartics, we can construct a real flat one-parameter family π̃ :X→ D(0) such that

• X is the double cover of CP 2 ×D(0) ramified along Q and X is isomorphic tothe hypersurface in CP (1, 1, 1, 2) × D(0) defined by the polynomial equationf(x, y, z) + εz4t2 = w2;

• X0 is the double cover of CP 2 ramified along Q̃. Depending on the real schemerealized by the pair (RP 2,RQ̃), the real part of X0 is homeomorphic either to⊔ji=1 S

2 t {pt} or to⊔j−1i=1 S

2 t∨2j=1 S

2, where {pt} is a point.

• the fibers π̃−1(t) := Xt are non-singular (k-sphere real) degree 2 del Pezzosurfaces (with either k = j or k = j + 1 depending on RQt), for t 6= 0 (and treal).

Now, performing the blow up Blp : X′ → X at the node p of X, we obtain a realflat one-parameter family π̃′ : X′ → D(0) such that Bl−1p (p) =: T is a real quadricsurface with real structure dependent on f(x, y, z) and ε, the fibers π̃′−1(t) := X ′tare non-singular (k-sphere real) del Pezzo surfaces of degree 2 (with either k = j ork = j + 1 depending on RXt), for t 6= 0 (and t real), and X ′0 is equal to the union oftwo real algebraic surfaces S ∪ T , where S and T are as described in (1)− (3).

Corollary 3.26. Let X ′0 be a real reducible surface equal to the union of two realalgebraic surfaces S ∪ T , where S and T are as described in (1) − (3) of Proposition3.25. Then, there exists a real flat one-parameter family π̃′ : X′ → D(0), whereD(0) ⊂ C is a real disk centered in 0, the fibers π̃′−1(t) := X ′t are non-singular (k-sphere real) del Pezzo surfaces of degree 2 (with k = j, resp. k = j + 1), for t 6= 0(and t real), and the central fiber is X ′0.

Proof. The anti-canonical system of S is the minimal resolution of the double coverof CP 2 ramified along a real algebraic quartic Q̃ with a real non-degenerate doublepoint q as only singularity and such that RQ̃ is homeomorphic to either

⊔ji=1 S

1t{q}or to

⊔j−1i=1 S

1 t∨2j=1 S

1, where 1 ≤ j ≤ 3. Applying the proof of Proposition 3.25 toQ̃, we prove the statement.

3.8.1 Intermediate constructions: Constructions on quadric surfaces andon j-sphere real 1-nodal del Pezzo pairs of degree 2

In Section 3.9, we use Theorem 3.24 to end the proof of Theorem 3.9, Proposition3.10 and to prove Proposition 3.12. In order to do that we need some intermediateconstructions. Therefore, the aim of this section is to

• construct real algebraic curves with prescribed topology and intersection with agiven real curve in the quadric ellipsoid (Proposition 3.28), resp. in the quadrichyperboloid (Proposition 3.30);

• construct real algebraic curves with prescribed topology on j-sphere real 1-nodaldegree 2 del Pezzo pairs (Proposition 3.32).

Notation 3.27. As mentioned in Section 2.1, the interior and exterior of any oval Din S2 is not well defined because it depends on a choice of a point in S2 \ D.In the proof of Propositions 3.28 and 3.29, whenever we talk about interior andexterior of an oval in S2 is with respect to a chosen point p ∈ S2 \ D and, to be clear

20

on the choice, we depict S2 projected from such p on a plane in Fig. 12 − 17. Recallthat every oval in S2 bounds two disks. Then, we say that the disk containing p isthe exterior, and the other one is the interior.

In the proofs of Propositions 3.28, 3.29, 3.30, we use variants of Harnack’s constructionmethod ([Har76]).

Proposition 3.28. Let T be the quadric ellipsoid and let ET be a non-singular realalgebraic curve of bidegree (1, 1) in T . Then, for any real configuration P2k of 2kdistinct points in ET fixed as follows, there exists a non-singular real algebraic curveCT of bidegree (k, k) on T , intersecting transversely ET in the 2k points and suchthat the triplet (RT,RET ,RCT ) is arranged respectively as depicted:

(1) in Fig. 12.1 for k = 2 and 4 fixed real points;

(2) in Fig. 12.2 for k = 3 and no fixed real points;

(3) in Fig. 12.3 for k = 3 and 2 fixed real points;

Fig. 12.1 Fig. 12.2

Fig. 12.3

Figure 12: RET in dashed.

Proof. For any real configuration P4 ⊂ RET let us construct a real curve H̃ of bidegree(2, 2) passing through P4 and such that the arrangement of RH̃ ∪RET is as depictedin Fig. 12.1. First of all, remind that for any 2 fixed distinct points on ET , thereexists a bidegree (1, 1) real algebraic curve passing through them.Let P0(x, y)P1(x, y) = 0 be a real polynomial equation defining the union of ET anda bidegree (1, 1) real curve H such that the points of P4 belong to one connectedcomponent E of RET \ RH. Let H1 and H2 be two bidegree (1, 1) real curves suchthat H1 ∪H2 contains P4. Replace the left side of the equation P0(x, y)P1(x, y) = 0with P0(x, y)P1(x, y) + εf1(x, y)f2(x, y), where fi(x, y) = 0 is an equation for Hi andε is a sufficient small real number. Up to a choice of the sign of ε, one constructsa small perturbation H̃ of ET ∪ H, where H̃ is a bidegree (2, 2) non-singular realcurve such that

⋃2i=1Hi ∩ ET = H̃ ∩ ET and the triplet (RT,RET ,RH̃) is arranged

as depicted in Fig. 12.1.Now, for any real configuration P6 ⊂ ET \ RET with no real points, respectivelywith exactly 2 real points, we want to construct real curves CT of bidegree (3, 3)passing through P6 and such that the arrangement of RCT ∪ RET is as depicted inFig. 12.2 on the left, respectively in Fig. 12.3. One can construct CT applying asmall perturbation to ET ∪ H̃, respectively to ET ∪ H̃2, where H̃2 is a real curve ofbidegree (2, 2) such that:

• the triplet (RT,RET ,RH̃2) is arranged as depicted in Fig. 12.1;

• the real points of P6 belong to an interior connected component, respectivelyan exterior connected component, of RET \ RH̃2; see Notation 3.27.

Let us end the proof constructing real curves CT of bidegree (3, 3)

21

• containing a given real configuration P ′ = {p1, p1, p2, p2, p3, p3} of points onET \ RET , where pi and pi are complex conjugated points;

• such that the arrangement of RCT ∪RET is as depicted in Fig. 12.2 respectivelyin the center and on the right.

Let Πi be a pencil of hyperplanes with base points pi and pi, with i = 1, 2, 3. Wewant to show that one can always construct a non-singular real algebraic curve CT ofbidegree (3, 3) as perturbation of the union of three hyperplanes respectively of Π1,Π2 and Π3 such that the arrangement of the triplet (RT,RET ,RCT ) is respectively asdepicted in Fig. 13.2. Namely, we prove that one can always find three hyperplanesrespectively of Π1, Π2 and Π3 whose union and real arrangement with respect to RETis respectively as in Fig. 13.1. First of all, remark that on each of the two connected

Fig. 13.1 Fig. 13.2Figure 13: RET in dashed.

components of RT \ RET , the real part of the real hyperplanes of the pencil Πi varyfrom a real point qi to RET , with i = 1, 2, 3. Moreover, the real points q1, q2 andq3 are distinct points. There exist two real hyperplanes Hj ⊂ Πj and Hk ⊂ Πk such

Fig. 14.1 Fig. 14.2 Fig. 14.3 Fig. 14.4

Fig. 14.5 Fig. 14.6 Fig. 14.7 Fig. 14.8

Figure 14: Hj and Hk in dashed and thick black, Hi in gray.

that

• RHj is tangent to RHk at a real point sjk;

• the point qi do not belong to the interior of RHj and RHk (see Notation 3.27);

• RHj ∪ RHk is as depicted in Fig. 14.1 (resp. in Fig. 14.5).

Pick the real hyperplane Hi ⊂ Πi passing through sjk. Then, the real part of Hj ∪Hk ∪Hi is as depicted in Fig. 14.2 (resp. in Fig. 14.6). It follows that there exists areal hyperplane of the pencil Πi whose real arrangement with respect to RHj ∪RHkis as depicted in Fig. 14.3 (resp. in Fig. 14.7). In conclusion, a small perturbation ofthe union of such three hyperplanes has real part as depicted in Fig. 14.4 (resp. inFig. 14.8).

Proposition 3.29. Let T be the quadric ellipsoid and let ET be a non-singular realalgebraic curve of bidegree (1, 1) in T . Then,

• for any integer k ≥ 5,

• for any real configuration P2k of 2k distinct points, whose exactly 2 are real, inET ,

• for any given non-negative integers k1, k2 such that k1 + k2 = k − 4

22

there exists a non-singular real algebraic curve CT of bidegree (k, k) on T , intersect-ing transversely ET in P2k and such that the triplet (RT,RET ,RCT ) is arranged asdepicted in Fig. 15.

}k2

}k1

Figure 15: RET in dashed. The pair (RT,RCT ) realizes 1tN(k1, 1)tN(k2, 1t 〈1〉).

Fig. 16.1: (RT,RET ,RĤ5) Fig. 16.2: (RT,RET ,RĤ5)

Figure 16: RET in dashed.

Fig. 17.1:(RT,RET ,RĤ2)

Fig. 17.2 Fig. 17.3:(RT,RET ,RĤ3)

Fig. 17.4 Fig. 17.5:(RT,RET ,RĤ4)

Figure 17: RET in dashed and {p1, p2} = {•, •}.

Proof. The first step is to prove the statement for k = 5. After we end the proofby induction on k. Let us introduce some notation. Let us call Ĥi any real curveof bidegree (i, i) such that only one oval of RĤi, denoted with Di, intersects RET .Moreover, we denote with F1, . . . ,Fj and F̃1, . . . , F̃j the connected components ofRET \ Di respectively in the interior and in the exterior of Di; see Notation 3.27.Fix P10 ⊂ ET and denote with p1, p2 the two real points of P10. We start with theconstruction of a real curve Ĥ5 of bidegree (5, 5) passing through P10 and such thatthe triplet (RT,RET ,RĤ5) is arranged respectively as depicted in Fig. 16.1 and 16.2.If one can construct a real curve Ĥ4 of bidegree (4, 4) such that

(1) RĤ4 ∩ RET = D4 ∩ RET consists of 2 points;

(2) {p1, p2} ⊂ F1 of D4;

(3) the triplet (RT,RET ,RĤ4) is arranged as depicted in Fig. 17.5

then, the real curve Ĥ5 exists as small perturbation of Ĥ4 ∪ ET ; see the proof ofProposition 3.28 for details on small perturbation method.Let us construct Ĥ4. Fix a configuration P4 of 4 points on RET such that {p1, p2}belong to the same connected components of RET \ P4. Via small perturbation, weconstruct a real curve Ĥ2 of bidegree (2, 2) such that

• the triplet (RT,RET ,RĤ2) is arranged as depicted in Fig. 17.1;

23

• RĤ2 ∩ RET = D2 ∩ RET = P4;

• {p1, p2} ⊂ F̃1 of D2.

Now, fix a real configuration P6 of 6 points on ET such that (see Fig. 17.2)

• exactly 4 points are real;

• one of the fixed real points belongs to Fi ⊂ D2, for i = 1, 2;

• two of the fixed real points are on F̃1 ⊂ D2 and p1, p2 belong to the sameconnected component of F̃1 \ D2.

One can construct a small perturbation Ĥ3 of ET ∪ Ĥ2, where Ĥ3 is a bidegree (3, 3)non-singular real curve passing through P6 and such that the triplet (RT,RET ,RĤ3)is arranged as depicted in Fig. 17.3. To end the construction of Ĥ4, fix 8 points on ETsuch that exactly 2 points are real and belong to two different connected componentsof F̃1\{p1, p2} ⊂ D3; see Fig. 17.4. One obtains Ĥ4 as small perturbation of Ĥ3∪ET .

Let us proceed by induction to end the proof. Assume that the statement holdfor k − 1. Now, for any given non-negative integers k̃1, k̃2 such that k̃1 + k̃2 = k − 4,let us construct a bidegree (k, k) real algebraic curve CT passing through a given realconfiguration P2k ⊂ ET , whose exactly 2 points are real and such that the triplet(RT,RET ,RCT ) is arranged as depicted in Fig. 15.By induction, for any choice of k̃1, k̃2 and P2k, there exists a bidegree (k − 1, k − 1)real algebraic curve Ĥk−1 such that

• Ĥk−1 passes through a given real configuration P2k−2 ⊂ ET ;

• the triplet (RT,RET ,RĤk−1) is arranged as depicted in Fig. 15, where ki = k̃iand kj = k̃j − 1 with {i, j} = {1, 2} and k̃j 6= 0;

• the connected component F1 ⊂ Dk−1 contains the 2 real points of P2k.

Finally, let P0(x, y)P1(x, y) = 0 be a real polynomial equation defining the unionof ET and Ĥk−1 in some local affine chart of T . Pick k real curves Li of bidegree(1, 1) such that

⋃ki=1 Li passes trhough P2k. Replace the left side of the equation

P0(x, y)P1(x, y) = 0 with P0(x, y)P1(x, y) + εf1(x, y) . . . fk(x, y), where fi(x, y) = 0 isan equation for Li and ε is a sufficient small real number. Up to a choice of the signof ε, one constructs a small perturbation CT of ET ∪ Ĥk−1 which is the wanted realcurve of bidegree (k, k).

Proposition 3.30. Let T be the quadric hyperboloid and let ET be a non-singularreal algebraic curve of bidegree (1, 1) in T . Then, for any real configuration of 2kdistinct points in ET fixed as follows, there exists a non-singular real algebraic curveCT of bidegree (k, k) on T , intersecting transversely ET in the 2k points and suchthat the triplet (RT,RET ,RCT ) is arranged respectively as depicted:

(1) in Fig. 18.1, 18.2 and 18.3 for k = 3 and 2 fixed real points;

(2) in Fig. 18.4 for k = 3 and no fixed real points;

(3) in Fig. 18.5 and 18.6 for k = 2 and 4 fixed real points.

Proof. For any configuration P4 of 4 fixed points on RET let us construct real curvesH̃ of bidegree (2, 2) passing through P and such that the arrangements of RH̃ ∪RETare respectively as depicted in Fig. 18.5 and 18.6. Let P0(x, y)P1(x, y) = 0 be a realpolynomial equation defining the union of ET and a bidegree (1, 1) real curve H suchthat the points of P4 belong to one connected component E of RET \ RH. Let H1and H2 be two bidegree (1, 1) real curves such that H1∪H2 contains P4. Replace theleft side of the equation P0(x, y)P1(x, y) = 0 with P0(x, y)P1(x, y) + εf1(x, y)f2(x, y),where fi(x, y) = 0 is an equation for Hi and ε is a sufficient small real number. Upto a choice of the sign of ε, one constructs a small perturbation H̃ of ET ∪H, whereH̃ is a bidegree (2, 2) non-singular real curve such that

⋃2i=1Hi ∩ET = H̃ ∩ET and

the triplet (RT,RET ,RH̃) is arranged respectively as depicted in Fig. 18.5 and 18.6.Analogously, via small perturbation method we can construct real algebraic curves ofbidegree (3, 3) as described in (2)− (3) and end the proof.

24

Fig. 18.1 Fig. 18.2 Fig. 18.3

Fig. 18.4 Fig. 18.5 Fig. 18.6

Figure 18: RT ' S1 × S1 is depicted as a cylinder such that the S1’s depicted indashed on the sides of each cylinder are identified and represent RET .

In order to accomplish some particular constructions in the proof of Proposition3.32, we need the following lemma.

Lemma 3.31. There exist real algebraic curves Q̃ and C respectively of degree 4 and3 in CP 2 with a unique real non-degenerate double singularity at a point q, such thatthe triplets (RP 2,RQ̃,RC) realize the real scheme depicted in Fig. 19 and Fig. 20.

Fig. 19.1 Fig. 19.2 Fig. 19.3

Fig. 19.4 Fig. 19.5

Figure 19:

Figure 20:

Proof. Let us realize the real schemes in Fig. 19. Then, thanks to an analogue con-struction method, one also realizes all real schemes in Fig. 20. In fact, remark that

25

Fig. 21.1 Fig. 21.2 Fig. 21.3

Fig. 21.4 Fig. 21.5

Figure 21:

one gets the real schemes in Fig. 20, just deleting the empty ovals of the real quarticschemes in Fig. 19.The blow-up of CP 2 at the point q is the first Hirzebruch surface Σ1 (Section 2.2).Then, in order to prove the statement, it is sufficient to construct reducible realalgebraic curves Ki, with i = 1, 2, 3, 4, 5, of bidegree (3, 4) in Σ1 as union of twonon-singular real algebraic curves Q and A respectively of bidegree (2, 2) and (1, 2)in Σ1 such that the pairs (RΣ1,RQ ∪ RA) realize the L-schemes in Fig. 21.Let us denote with η̃i the trigonal L-schemes in RΣ5 respectively depicted in Fig.

Fig. 22.1

p1

p3p4 p2

Fig. 22.2

p1

p3p4 p2

Fig. 22.3

p1

p3p4 p2

Fig. 22.4

p1

p3

p4p2

Fig. 22.5

p1

p3 p4

p2

Figure 22: Intermediate constructions.

Fig. 23.1 Fig. 23.2 Fig. 23.3 Fig. 23.4

Figure 23: Intermediate constructions.

22, for i = 1, 2, 3, 4, 5. Due to Theorem 2.9, if the real graph associated to each η̃iis completable in degree 5 to a real trigonal graph, then there exists a real algebraictrigonal curve K̃i realizing η̃i, for all i ∈ {1, 2, 3, 4, 5}. Therefore, the completion Γiof the real graph, associated to each η̃i, respectively depicted in Fig. 23.1 - 23.3 andFig. 23.4 for i = 1, 2, 3 and i = 4, 5, proves the existence of such K̃i’s.Each K̃i is reducible because it has 12 non-degenerate double points and its normal-ization has 4 real connected components. In particular, every K̃i has to be the union

26

of a real curve of bidegree (2, 0) and a real curve of bidegree (1, 0).Let us consider the birational transformation

Ξ := β−1p1 β−1p2 β

−1p3 β

−1p4 : (Σ5, K̃i) 99K (Σ1,Ki),

defined as in Section 2.2; where the points pj ’s, with j = 1, 2, 3, 4, are the real doublepoints of K̃i respectively as depicted in Fig. 22, and the dashed real fibers are thoseintersecting the pj ’s. The image via Ξ of the reducible real trigonal curve K̃i is areducible curve Ki of bidegree (3, 4) which is the union of two non-singular real curvesQ and A, respectively of bidegree (2, 2) and (1, 2) in Σ1. Moreover, the L-scheme ofeach Ki is as respectively depicted in Fig. 21.

We end this section giving the following intermediate constructions.

Proposition 3.32.

(i) For every arrangement of ovals in⊔si=1 S

2 respectively depicted in Fig. 24, Fig.25 and Fig. 26, for every s ≤ j ≤ 3, there exists a j-sphere real 1-nodal delPezzo pair (S,ES), and real algebraic curve CS ⊂ S of bi-class (d, k) such thatRCS ∪ RES is arranged in RS as depicted

(1) in Fig. 24, for d = 3 and k = 2;

(2) in Fig. 25 and in Fig. 26, for d = 3 and k = 3.

(ii) Let d, k1, k2 and h1, h2, h3, h4 be non-negative integers such that

• d ≥ 5;• k1 + k2 = d− 4;

•4∑i=1

hi = d− 1;

• h3 ≡ 1 mod 2, if h3 6= 0;• h4 ≡ 1 mod 2, if h4 6= 0.

Then, there exist a 3-sphere real 1-nodal del Pezzo pair (S,ES) and a bi-class(d, d) real curve CS ⊂ S such that RCS ∪RES is arranged in RS as depicted inFig. 27.

Fig. 24.1:s = 1

Fig. 24.2:s = 1

Fig. 24.3:s = 2

Fig. 24.4:s = 1

Fig. 24.5:s = 1

Figure 24: RES ' S1 in dashed.

Proof. Let Q̃ be a real quartic with a real non-degenerate and non-isolated doublepoint at q as only singularity and let C be a real curve of degree d with one k-foldsingularity at q. To a pair (Q̃, C) correspond a real 1-nodal del Pezzo pair (S,ES)and a real algebraic curve CS ⊂ S of bi-class (d, k) with topology described by thetopological type realized by the triplet (RP 2,RQ̃,RC).Proof of (i): First of all, Lemma 3.31 immediately implies the existence of j-spherereal 1-nodal degree 2 del Pezzo pairs (S,ES) and real curves CS ⊂ S of bi-class (3, 2)such that the triplet (RS,RES ,RCS) is arranged as depicted in Fig. 24, where wedepict only the non-empty spheres of RS.Let us realize the arrangements in (2). It is easy to see that there exist a real planequartic Q̃1 with a real non-degenerate double point q1 as only singularity and realpart homeomorphic to

⊔j−1i=1 S

1 t∨2j=1 S

1 and a pencil of lines Lq1 ⊂ CP 2, centered

27



Fig. 27.1

}h1

}h

2

}

h3

} h4

Fig. 27.2

}

h3

}

h4

}h2

}h1

Figure 27: RES ' S1 in dashed. The pair (RS,RCS) realizes 〈N(h1, 0)〉 tN(h2, 0) :N(h3, 0) : N(h4, 0).

Fig. 28.1: (RP 2,RLq1 ,RQ) Fig. 28.2

RL RL

Figure 28: 1: The set of points of RQ homeomorphic to⊔j−1i=1 S

1 is depicted in dashed.2: The lines in dashed represent the d− 1 lines of the pencil Lq1 .

28

at q1, such that RQ̃1 ∪ RLq1 is arranged respectively as depicted in Fig. 28.1. Theunion of any three distinct lines of Lq1 is a cubic C1 with a triple point at q1. FromQ̃1 and C1 one can construct j-sphere real 1-nodal degree 2 del Pezzo pairs (S,ES)and curves CS ⊂ S of bi-class (3, 3) such that the triplet (RS,RES ,RCS) is arrangedas depicted in Fig. 25 (resp. Fig. 26) where we depict only the non-empty spheres ofRS.Proof of (ii): Assume that Q̃1 has real part homeomorphic to

⊔2i=1 S

1 t∨2j=1 S

1.The union of a line L ⊂ Lq1 (in thick black) and other d− 1 distinct lines (in dashed)of Lq1 respectively as depicted in Fig. 28.2 is a degree d real curve Cd with a d-foldsingularity at q1. From Q̃1 and Cd one can construct 3-sphere real 1-nodal degree 2del Pezzo pairs (S,ES) and real curves CS ⊂ S of bi-class (d, d) such that the triplet(RS,RES ,RCS) is arranged respectively as depicted in Fig. 27.1 and 27.2.

3.9 Final constructionsWe end the proof of Theorem 3.9 and Proposition 3.10. Moreover, we prove Propo-sition 3.12. The proofs combine the results and constructions of Theorem 3.24 andPropositions 3.28, 3.29, 3.30, 3.32.

Proposition 3.33. Every real scheme S in SDP2(4, 3) labeled with † in Table 3, isrealizable in X4 and in class 3. Moreover, every S labeled with †∗ is realizable in Xkand in class 3, with 1 ≤ k ≤ 3.

Proof. The realization of real schemes in class 3 is done as follows.General construction: Pick any j-sphere real 1-nodal degree 2 del Pezzo pair(S,ES) and any real algebraic curve CS ⊂ S of bi-class (3, h) constructed as in proofof Proposition 3.32, with h = 2, 3. Due to Corollary 3.26, there exists a real algebraicsurface X ′0 as union of S and T , intersecting along a curve E, and there exists a realalgebraic curve C0 as union of CS and CT , intersecting along 2h points of E; whereT is a quadric ellipsoid, respectively a quadric hyperboloid and CT ⊂ T is a real alge-braic curve of bidegree (h, h) constructed as in proof of Proposition 3.28, respectivelyProposition 3.30. Then, thanks to Theorem 3.24, one realizes a real scheme in Xj+1,respectively in Xj and in class 3.

Applying the above general construction in 4 different ways, one realizes in Xkand in class 3 all the real schemes listed below. Let us divides such real schemes in 4groups:

(1) for 1 ≤ k ≤ 4,

1 t 〈1〉 t 〈1〉 : 3 : 0 : 0,2 t 〈2〉 : 3 : 0 : 0,

1 t 〈3〉 : 〈〈1〉〉 : 0 : 0,1 t 〈2〉 : 1 t 〈2〉 : 0 : 0,

5 : 3 : 0 : 0, 1 t 〈2〉 : 2 : 2 : 0,4 : 4 : 0 : 0

(2) for 1 ≤ k ≤ 3

1 t 〈1〉 t 〈2 t 〈1〉〉 : 0 : 0 : 0,1 t 〈1〉 t 〈1〉 t 〈2〉 : 0 : 0 : 0,4 t 〈1〉 t 〈1〉 : 0 : 0 : 0,

1 t 〈2〉 t 〈3〉 : 0 : 0 : 0,3 t 〈4〉 : 0 : 0 : 0,8 : 0 : 0 : 0

(3) for 1 ≤ k ≤ 3

2 t 〈5〉 : 0 : 0,3 t 〈1〉 t 〈2〉 : 0 : 0,1 t 〈3〉 t 〈〈1〉〉 : 0 : 0,

1 t 〈1〉 t 〈4〉 : 0 : 0,1 t 〈〈4〉〉 : 0 : 0,2 t 〈2〉 t 〈〈1〉〉 : 0 : 0,

2 t 〈1〉 t 〈3〉 : 0 : 0,1 t 〈1〉 t 〈〈3〉〉 : 0 : 0,3 t 〈1〉 t 〈〈1〉〉 : 0 : 0

(4) for 1 ≤ k ≤ 4, all the remaining real schemes labeled with † and/or †∗ in Table3.

Now let us apply the general construction to each case as follows.

(1) Take j+1 = k and h = 2. Let T be a quadric ellipsoid and CT ⊂ T a real curveof bidegree (2, 2) constructed as in proof of Proposition 3.28.

29

(2) Take j = k and h = 2. Let T be a quadric hyperboloid and CT ⊂ T a real curveof bidegree (2, 2) constructed as in proof of Proposition 3.30. Moreover, takeCS ⊂ S such that the triplet (RS,RES ,RCS) is as depicted respectively in Fig.24.1, 24.2, 24.4 and 24.5, where we depict only the non-empty spheres of RS.

(3) Take j = k, h = 3: Let T be a quadric hyperboloid and CT ⊂ T constructedas in proof of Proposition 3.30. Moreover, take CS ⊂ S such that the triplet(RS,RES ,RCS) is as depicted in Fig. 29.

(4) Take j + 1 = k, h = 3. Let T be a quadric ellipsoid and CT ⊂ T constructed asin proof of Proposition 3.28.


Example 3.34. We follow the steps of the proof of Proposition 3.33 to realize:

(1) the real scheme 〈1〉 t 〈2〉 : 3 : 0 : 0 in X4 and in class 3.

• Let CS ⊂ S be the real algebraic curve such that RES ∪ RCS is arrangedin RS as pictured in Fig. 30.1.• Let T be the quadric ellipsoid and let CT ⊂ T be the real algebraic curve

of bidegree (3, 3) such that RET ∪ RCT is arranged in RT as depicted inFig. 30.2.

Fig. 30.1 Fig. 30.2 Fig. 30.3

Figure 30: RE ' S1 in thick dashed.

• Thanks to Theorem 3.24 〈1〉 t 〈2〉 : 3 : 0 : 0 is realizable in X4 and in class3 (Fig. 30.3);

(2) the real scheme bt 〈a+ 1〉 t 〈〈1〉〉 : 0 : 0 in Xk and in class 3, where a, b denotesnumber of ovals and a+ b = 3 and k = 3, 2, 1.

• Let CS ⊂ S be the real algebraic curve such that RES∪RCS is arranged inRS as pictured in Fig. 31.1, where we depict only the non-empty spheresof RS.• Let T be the quadric hyperboloid and CT ⊂ T be the real algebraic curve

of bidegree (3, 3) such that RET ∪ RCT is arranged in RT as depicted inFig. 31.2.

• Thanks to Theorem 3.24, for any values of a, b, the real scheme b t 〈a +1〉 t 〈〈1〉〉 : 0 : 0 : 0 is realizable in Xk and in class 3. See Fig. 31.3, wherewe depict only the non-empty spheres of RX.

30

Fig. 31.1 Fig. 31.2

a

b

Fig. 31.3

a

b

Figure 31: RE ' S1 in thick dashed.

The following definition is used as a (non-)symmetry detector in the proof ofProposition 3.12.

Definition 3.35. Let S be a topological type in S2. We say that S has a mirrorif there exist an element in the equivalence class of S (Definition 2.2) of the formS̃ t S̃ t T , with S̃ different from 0. Otherwise, we say that S has no mirrors.

Example 3.36. Let S be the topological type 1t〈〈1〉〉 in S2. There ex

Real algebraic curves on real del Pezzo...

Documents

Transcript of Real algebraic curves on real del Pezzo...